Optimisation

% A workshop makes tables (profit 30) and chairs (profit 20).
% Hours available: carpentry 120 (tables 4 h, chairs 2 h), finishing 90 (tables 2 h, chairs 3 h).
% At least 5 chairs must be made. linprog minimises, so we minimise minus the profit.
f = -[30; 20];
A = [4 2; 2 3; 0 -1];  b = [120; 90; -5];
[x, fval] = linprog(f, A, b, [], [], [0; 0]);

[T, Cc] = meshgrid(0:0.25:35, 0:0.25:35);
ok = 4*T + 2*Cc <= 120 & 2*T + 3*Cc <= 90 & Cc >= 5;
contourf(T, Cc, 30*T + 20*Cc .* ok - 1e3*~ok, [0 200:200:1200]), colormap(gray), hold on
t = 0:35;
plot(t, (120 - 4*t)/2, t, (90 - 2*t)/3, 'LineWidth', 1.8), yline(5, '--')
plot(x(1), x(2), 'rp', 'MarkerSize', 14), hold off
xlim([0 35]), ylim([0 35]), xlabel('tables'), ylabel('chairs')
legend('profit (feasible area)', 'carpentry hours', 'finishing hours', 'minimum chairs', 'optimum')
title(sprintf('Best plan: %.1f tables, %.1f chairs, profit %.0f', x(1), x(2), -fval))

% The Rosenbrock "banana" function, restricted to the unit disk (MATLAB's fmincon example)
f = @(x) 100*(x(2) - x(1)^2)^2 + (1 - x(1))^2;
disk = @(x) deal(x(1)^2 + x(2)^2 - 1, []);        % c(x) <= 0, no equality constraints
[x, fval] = fmincon(f, [0; 0], [], [], [], [], [], [], disk);
xu = fminunc(f, [-1; 2]);                         % without the constraint

[X, Y] = meshgrid(linspace(-1.5, 1.5, 120), linspace(-0.6, 1.6, 100));
contour(X, Y, log(1 + 100*(Y - X.^2).^2 + (1 - X).^2), 20), hold on
th = linspace(0, 2*pi); plot(cos(th), sin(th), 'k', 'LineWidth', 1.5)
plot(x(1), x(2), 'rp', xu(1), xu(2), 'bo', 'MarkerSize', 10), hold off
axis equal, legend('log(1 + f)', 'constraint x² + y² ≤ 1', 'fmincon', 'fminunc (no constraint)')
title(sprintf('Constrained minimum (%.4f, %.4f), f = %.4f', x(1), x(2), fval))

% Enzyme kinetics: Michaelis–Menten v = Vmax S / (Km + S), fitted to simulated measurements
rng(8)
S = [0.2 0.5 1 2 4 8 15 30];
v = 10 * S ./ (2.5 + S) .* (1 + 0.05*randn(size(S)));
model = @(p, S) p(1) * S ./ (p(2) + S);
[p, resnorm, residual] = lsqcurvefit(model, [5; 1], S, v, [0; 0], []);   % parameters stay positive
Sf = linspace(0, 32, 200);
subplot(2, 1, 1)
plot(S, v, 'o', Sf, model(p, Sf), 'LineWidth', 1.8)
xlabel('substrate S (mM)'), ylabel('rate v'), legend('measured', 'fit')
title(sprintf('Vmax = %.2f, Km = %.2f  (true: 10 and 2.5)', p(1), p(2)))
subplot(2, 1, 2)
stem(S, residual), yline(0), xlabel('S (mM)'), title('residuals')

% Choose items (take it or not) to maximise value within a 15 kg limit
names  = {'tent', 'stove', 'food', 'water', 'camera', 'book', 'jacket', 'map'};
weight = [5 2 4 6 1 1 3 0.2];
value  = [9 4 8 9 3 1 5 2];
n = numel(value);
[x, fval] = intlinprog(-value', 1:n, weight, 15, [], [], zeros(n, 1), ones(n, 1));
barh(value .* x'), yticklabels(names), xlabel('value packed')
title(sprintf('Packed value %d with %.1f kg of 15', -fval, weight * x))